Persistence of Freeness for Lie Pseudogroup Actions

The action of a Lie pseudogroup $G$ on a smooth manifold $M$ induces a prolonged pseudogroup action on the jet spaces $J^n$ of submanifolds of $M$. We prove in this paper that both the local and global freeness of the action of $G$ on $J^n$ persist under prolongation in the jet order $n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the pseudogroup action on $J^\infty$.